On Fri, 25 Feb 2000, JoeShipman at aol.com wrote:
> 1) Vopenka's Principle is one of my favorite large cardinal axioms,
> since it is better motivated and easier to state than almost all of
> them (inaccessibles and measurables are easier still, but I can't
> think of any others that are easier).
>> 2) This is a use of large cardinals to settle an open problem in "ordinary
> mathematics" (excluding logic, foundations, set theory). Have
> there been any other good examples of this since Solovay's work on
> real-valued measurables?
I don't know if this is exactly what Shipman was looking for, but,
since Vop^enka's principle is one of his favorites, he might be
interested in the book
Ji^r'i Ad'amek, Ji^i'i Rosick'y, Locally Presentable and
Accessible Categories, LMS Lecture Notes Series 189, Cambridge
University Press, 1994,
especially
Chapter 6 (40 pages): Vop^enka's Principle
Appendix (14 pages): Large Cardinals
and the list of 16 open problems with which the book ends, 12 of
which are settled in the affirmative using Vop^enka's Principle (or
the weak Vop^enka Principle).
The subject matter of the book is a very general but yet very useful
class of categories that includes all of the usual algebraic and
semi-algebraic categories (the varieties and quasi-varieties of
universal algebra) and all Horn classes of relational structures.
(I apologize for my ASCII attempts at the ^Cech diacritics; I'm not
Unicode-enabled yet.)
--
Todd Wilson
Computer Science Department
California State University, Fresno